1. Consider a Dirichlet series
PI
n
=1
Exercise 1, 9/08/2005
an n s .
(a) If the series converges for s = s0 , then it converges uniformly in any angular region
n
s
P C : +
2
arg(s s0 )
2
o
dened by a real number such that 0 < < .
2
(b) According to (a)
Exercise 10, 11/20/2005
1. Let F be a number eld. Find an explicit bound, C , in terms of jdisc(F=Q)j, and rI ; rP for
F , such that every element of the ideal class group for F is represented by an ideal I & yF
such that jyF =I j C . Illustrate this meth
Exercise 9, 11/10/2005
1. Determine explicitly the Pontriagin dual of Zp and Qp .
2. Determine the Pontiagin dual of Z, where Z is the pronite completion of Z.
3. Identify C with its own Pontriagin dual via
(z; w) U3 exp 2
p
1TrC=R(zw)
Determine the self-
Exercise 8, 10/25/2005
1. Let F be a global eld.
(i) Is the set
(
x
a closed subset of
= (x ) P A :
v
Y
F
jj
xv
)
v
=0
v
A
F
?
(ii) Let f (t) be a polynomial over F . Let
(
x
a>
0 be a positive real number. Is the set
= (x ) P A :
v
Y
F
j
( )j T= a
f xv
)
Exercise 7, 10/23/2005
p
1. Let G = Gal(Q(3 ; 3 2)=Q). Determine the upper and lower numbering ltration of the
decompostion subgroup of G above p = 2 and p = 3.
2. Let G = Gal Qp (p2 )=Qp , where p is an odd prime number. Determine the upper and
lower num
Exercise 6, 10/05/2005
Denition Let L/K be a nite separable extension eld. The discriminant of a K -basis
1 , . . . , n of L is dened to be
d(1 , . . . , n ) := det (i (j )2
where 1 , . . . , n are the K -embeddings of L into Lsep . If is an element of L
Exercise 5, 10/03/2005
1. Determine explicitly the left invariant Haar measure on the locally compact group GL2 (Qp ).
2. Let f (x) be the characteristic function on Zp . Consider the integral
Z
f (x)jjxjjs dx
sPC
Qp
Determine all s P C for which the abov
Exercise 3a, 10/9/2005
Correction to Exercise 3 and discussion of related issues
1. Comments on Problem 3, Exercise set 3.
(i) Statements (i), (ii) of are wrong. In fact the abscissa of convergence for F (s) is at most
1 ; see Problem 3 below.
2
(ii) Ther
Exercise 3, 9/18/2005
This short set of problems provides a counter-example for Question 3 in Exercise 1.
1. Let
P
s
n!1 an n be a Dirichlet series which is divergent for s = 0.
Prove that the abscissa
of convergence of this Dirichlet series is equal to
l
ixerise PeD WGPRGPHHS
rere is some hints for rolem I in ixerise set PF vet e nonEtrivil hirihlet hrterF
IF rove tht
0
1
@m A
@nA
log n a [email protected]; A @
@mAA
n
m
m x
nx
X
X
where
X
R1 @x; A a
nd
PF how tht
n>x
R2 @x; A a
@nA
n
!
R @x; A R @x; A ;
1
0
2
1
@mA
@
ixerise PD WGIPGPHHS
IF vet e nonEprinipl hirihlet series modulo nD n P N!3 F how tht
[email protected]; A a
X @nA
n!1
a
n
Y
p
@p A 1
I p
PF vet d e positive integerF pind formul for the numer of primitive hirihlet hrters
modulo d in terms of the prime ftoriztion of d
ixerise IHD IPGHTGPHHS
IF vet F e lol eldD nd denote y @; ; A the lol Eftor tthed to qusiE
hrter of F D nonEtrivil hrter of @F; CAD nd rr mesure for @F; CAF
rove tht
@iA @;
; cA
@iiA @; x U3
a c @;
; A
for every c > HF
@axA; A a @aA jaj1 @; ; A for every